Hi Janelle,

Consider an adjacency matrix as follows:

A B C D
0 1 0 1
1 0 1 1
0 1 0 0
1 1 0 0

You could then calculate a distance matrix which would calculate the number of edges you would have to pass through in order to go from one to another:


A B C D
0 1 2 1
1 0 1 1
2 1 0 2
1 1 2 0

You can interpret it as follows:

In order to walk from A to B, you need to pass through 1 edge
In order to walk from A to C, you need to pass through 2 edges (because A and C are not directly connected; you would have to walk via B)
In order to walk from A to D, you need to pass through 1 edge
and so on.

So in a binary network, you could think of it as the nodes being connected directly or indirectly. If you go by the assumption that information flows via shortest routes in a network, then shortest path lengths could be a useful index to quantify how information is flowing in a network. There are studies which have shown that in certain disorders, the mean shortest path length (also known as characteristic path length) are increased as compared to healthy brains. For a weighted network, you would typically transform the weighted adjacency matrix into a connection matrix having some sort of mapping between correlation values and lengths. For example, an inverse mapping such that higher correlation values would be shown as shorter paths. The interpretation of path lengths in a weighted network is rather ambiguous. You could still use the hop counting method (as in binarized case) to calculate path lengths or else sum the edge weights. I suggest having a look at Chapter 7 of 'Fundamentals of Brain Network Analysis' by Fornito, Zalesky, and Bullmore for thoughts on interpreting path lengths for weighted and unweighted network.

A value of 0 and 1 in your adjacency matrix would mean binarized graph. And indeed, a value of 1 would mean a direct connection between the two nodes while a value of 0 would mean that a direct connection between the two nodes is absent. Since you are working with functional data rather than anatomical data, you should think of them as functional connections (for example, two brain regions which may or may not be anatomically directly connected will work together when performing a task). When calculating path length in such a case, a path length of 1 would mean that the areas are directly connected and a value more than 1 would mean that the areas are indirectly connected (consequently you have to move through several more brain areas in order to reach the destination; see example above where A and C are not directly connected and thus have a higher path length). Functionally speaking, you could think of them as exchanging information directly or indirectly. For example, A and C need to communicate indirectly because the output of A needs to be further processed by B before it is sent to C. As a crude analogy, think of A as the input device of a computer, B as the processing unit, and C as the output unit. So A and C are indirectly (functionally) connected because information from A can only go to C after B has processed it. Depending on the nature of the task being performed, it might make sense to think of the brain regions doing some segregated processing. Similarly, it may make sense to think of processing happening in modules which are connected by paths. 


Hope the above helps.



Best
Pravesh